p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.447D4, C42.332C23, C4○D4⋊3Q8, C4○(D4.Q8), C4○(Q8.Q8), Q8.2(C2×Q8), D4.2(C2×Q8), C4○(D4⋊2Q8), C4○(D4⋊Q8), C4○(Q8⋊Q8), C4○(C4.Q16), Q8.Q8⋊53C2, D4.Q8⋊53C2, D4⋊Q8⋊48C2, D4⋊2Q8⋊46C2, C4.Q16⋊48C2, Q8⋊Q8⋊48C2, C4⋊C4.39C23, C4.111(C4○D8), C4.27(C22×Q8), C4⋊C8.283C22, (C2×C4).274C24, (C2×C8).139C23, C23.386(C2×D4), (C22×C4).431D4, C4⋊Q8.260C22, (C2×D4).393C23, (C4×D4).315C22, C23.25D4⋊4C2, C4.124(C22⋊Q8), (C2×Q8).364C23, (C4×Q8).296C22, C2.D8.166C22, C4.Q8.147C22, (C22×C8).180C22, (C2×C42).820C22, C23.24D4.7C2, C22.534(C22×D4), C22.19(C22⋊Q8), D4⋊C4.158C22, C2.17(D8⋊C22), C23.37C23⋊2C2, (C22×C4).1544C23, Q8⋊C4.149C22, C42.C2.102C22, C42⋊C2.115C22, (C2×C4⋊C8)⋊29C2, C2.19(C2×C4○D8), C4.84(C2×C4○D4), (C2×C4)○(Q8.Q8), (C2×C4)○(Q8⋊Q8), (C2×C4)○(C4.Q16), (C4×C4○D4).23C2, (C2×C4).322(C2×Q8), C2.55(C2×C22⋊Q8), (C2×C4).1435(C2×D4), (C2×C4).840(C4○D4), (C2×C4○D4).307C22, SmallGroup(128,1808)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.447D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c3 >
Subgroups: 324 in 194 conjugacy classes, 102 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22×C8, C2×C4○D4, C23.24D4, C2×C4⋊C8, C23.25D4, D4⋊Q8, Q8⋊Q8, D4⋊2Q8, C4.Q16, D4.Q8, Q8.Q8, C4×C4○D4, C23.37C23, C42.447D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C4○D8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C4○D8, D8⋊C22, C42.447D4
►On 64 points
(1 55 41 63)(2 64 42 56)(3 49 43 57)(4 58 44 50)(5 51 45 59)(6 60 46 52)(7 53 47 61)(8 62 48 54)(9 27 36 21)(10 22 37 28)(11 29 38 23)(12 24 39 30)(13 31 40 17)(14 18 33 32)(15 25 34 19)(16 20 35 26)
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 52 13 56)(10 53 14 49)(11 54 15 50)(12 55 16 51)(17 42 21 46)(18 43 22 47)(19 44 23 48)(20 45 24 41)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 48 41 8)(2 7 42 47)(3 46 43 6)(4 5 44 45)(9 14 36 33)(10 40 37 13)(11 12 38 39)(15 16 34 35)(17 22 31 28)(18 27 32 21)(19 20 25 26)(23 24 29 30)(49 60 57 52)(50 51 58 59)(53 64 61 56)(54 55 62 63)
G:=sub<Sym(64)| (1,55,41,63)(2,64,42,56)(3,49,43,57)(4,58,44,50)(5,51,45,59)(6,60,46,52)(7,53,47,61)(8,62,48,54)(9,27,36,21)(10,22,37,28)(11,29,38,23)(12,24,39,30)(13,31,40,17)(14,18,33,32)(15,25,34,19)(16,20,35,26), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,52,13,56)(10,53,14,49)(11,54,15,50)(12,55,16,51)(17,42,21,46)(18,43,22,47)(19,44,23,48)(20,45,24,41)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,48,41,8)(2,7,42,47)(3,46,43,6)(4,5,44,45)(9,14,36,33)(10,40,37,13)(11,12,38,39)(15,16,34,35)(17,22,31,28)(18,27,32,21)(19,20,25,26)(23,24,29,30)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63)>;
G:=Group( (1,55,41,63)(2,64,42,56)(3,49,43,57)(4,58,44,50)(5,51,45,59)(6,60,46,52)(7,53,47,61)(8,62,48,54)(9,27,36,21)(10,22,37,28)(11,29,38,23)(12,24,39,30)(13,31,40,17)(14,18,33,32)(15,25,34,19)(16,20,35,26), (1,26,5,30)(2,27,6,31)(3,28,7,32)(4,29,8,25)(9,52,13,56)(10,53,14,49)(11,54,15,50)(12,55,16,51)(17,42,21,46)(18,43,22,47)(19,44,23,48)(20,45,24,41)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,48,41,8)(2,7,42,47)(3,46,43,6)(4,5,44,45)(9,14,36,33)(10,40,37,13)(11,12,38,39)(15,16,34,35)(17,22,31,28)(18,27,32,21)(19,20,25,26)(23,24,29,30)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63) );
G=PermutationGroup([[(1,55,41,63),(2,64,42,56),(3,49,43,57),(4,58,44,50),(5,51,45,59),(6,60,46,52),(7,53,47,61),(8,62,48,54),(9,27,36,21),(10,22,37,28),(11,29,38,23),(12,24,39,30),(13,31,40,17),(14,18,33,32),(15,25,34,19),(16,20,35,26)], [(1,26,5,30),(2,27,6,31),(3,28,7,32),(4,29,8,25),(9,52,13,56),(10,53,14,49),(11,54,15,50),(12,55,16,51),(17,42,21,46),(18,43,22,47),(19,44,23,48),(20,45,24,41),(33,57,37,61),(34,58,38,62),(35,59,39,63),(36,60,40,64)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,48,41,8),(2,7,42,47),(3,46,43,6),(4,5,44,45),(9,14,36,33),(10,40,37,13),(11,12,38,39),(15,16,34,35),(17,22,31,28),(18,27,32,21),(19,20,25,26),(23,24,29,30),(49,60,57,52),(50,51,58,59),(53,64,61,56),(54,55,62,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | D8⋊C22 |
| kernel | C42.447D4 | C23.24D4 | C2×C4⋊C8 | C23.25D4 | D4⋊Q8 | Q8⋊Q8 | D4⋊2Q8 | C4.Q16 | D4.Q8 | Q8.Q8 | C4×C4○D4 | C23.37C23 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.447D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,10,16,0,0,16,7],[3,14,0,0,14,14,0,0,0,0,10,16,0,0,16,7] >;
C42.447D4 in GAP, Magma, Sage, TeX
C_4^2._{447}D_4 % in TeX
G:=Group("C4^2.447D4"); // GroupNames label
G:=SmallGroup(128,1808);
// by ID
G=gap.SmallGroup(128,1808);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations